Variational problems on graphs and their continuum limits
Dejan Slepcev, Carnegie Mellon University

Abstract: We will discuss variational problems arising in machine learning and their limits as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying "ground-truth" measure. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points.
Many machine learning tasks, such as clustering and classification, can be posed as minimizing  functionals on such graphs. We consider functionals involving graph cuts and graph laplacians and their limits as the number of data points goes to infinity.  In particular we establish under what conditions the minimizers of discrete problems have a well defined continuum limit, and characterize the limit.
The talk is primarily based on joint work with Nicolas Garcia Trillos, as well as on works with Xavier Bresson, Moritz Gerlach, Matthias Hein, Thomas Laurent, James von Brecht and Matt Thorpe.